Group action invariants
Degree $n$ : | $7$ | |
Transitive number $t$ : | $5$ | |
Group : | $\GL(3,2)$ | |
CHM label : | $L(7) = L(3,2)$ | |
Parity: | $1$ | |
Primitive: | Yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
Generators: | (1,2)(3,6), (1,2,3,4,5,6,7) | |
$|\Aut(F/K)|$: | $1$ |
Low degree resolvents
NoneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
7T5, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy Classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 1, 1, 1 $ | $21$ | $2$ | $(3,5)(6,7)$ |
$ 4, 2, 1 $ | $42$ | $4$ | $(2,3,4,7)(5,6)$ |
$ 3, 3, 1 $ | $56$ | $3$ | $(2,3,5)(4,7,6)$ |
$ 7 $ | $24$ | $7$ | $(1,2,3,4,5,6,7)$ |
$ 7 $ | $24$ | $7$ | $(1,2,3,7,6,4,5)$ |
Group invariants
Order: | $168=2^{3} \cdot 3 \cdot 7$ | |
Cyclic: | No | |
Abelian: | No | |
Solvable: | No | |
GAP id: | [168, 42] |
Character table: |
2 3 3 2 . . . 3 1 . . 1 . . 7 1 . . . 1 1 1a 2a 4a 3a 7a 7b 2P 1a 1a 2a 3a 7a 7b 3P 1a 2a 4a 1a 7b 7a 5P 1a 2a 4a 3a 7b 7a 7P 1a 2a 4a 3a 1a 1a X.1 1 1 1 1 1 1 X.2 3 -1 1 . A /A X.3 3 -1 1 . /A A X.4 6 2 . . -1 -1 X.5 7 -1 -1 1 . . X.6 8 . . -1 1 1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |